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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Superior highly composite number ： ウィキペディア英語版 Superior highly composite number In mathematics, a superior highly composite number is a natural number which has more divisors than any other number . It is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer. The first 10 superior highly composite numbers and their factorization are listed. For a superior highly composite number ''n'' there exists a positive real number ''ε'' such that for all natural numbers ''k'' smaller than ''n'' we have :$\backslash frac\backslash geq\backslash frac$ and for all natural numbers ''k'' larger than ''n'' we have :$\backslash frac>\backslash frac$ where ''d(n)'', the divisor function, denotes the number of divisors of ''n''. The term was coined by Ramanujan (1915). The first 15 superior highly composite numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 are also the first 15 colossally abundant numbers, which meet a similar condition based on the sum-of-divisors function rather than the number of divisors. == Properties == All superior highly composite numbers are highly composite. An effective construction of the set of all superior highly composite numbers is given by the following monotonic mapping from the positive real numbers.〔Ramanujan (1915); see also URL http://wwwhomes.uni-bielefeld.de/achim/hcn.dvi〕 Let :$e\_p(x)\; =\; \backslash left\backslash lfloor\; \backslash frac\; \backslash right\backslash rfloor\backslash quad$ for any prime number ''p'' and positive real ''x''. Then :$\backslash quad\; s(x)\; =\; \backslash prod\_\backslash quad$ is a superior highly composite number. Note that the product need not be computed indefinitely, because if $p\; >\; 2^x$ then $e\_p(x)\; =\; 0$, so the product to calculate $s(x)$ can be terminated once $p\; \backslash ge\; 2^x$. Also note that in the definition of $e\_p(x)$, $1/x$ is analogous to $\backslash varepsilon$ in the implicit definition of a superior highly composite number. Moreover for each superior highly composite number $s^\backslash prime$ exists a half-open interval $I\; \backslash subset\; \backslash R^+$ such that $\backslash forall\; x\; \backslash in\; I:\; s(x)\; =\; s^\backslash prime$ . This representation implies that there exist an infinite sequence of $\backslash pi\_1,\; \backslash pi\_2,\; \backslash ldots\; \backslash in\; \backslash mathbb$ such that for the ''n''-th superior highly composite number $s\_n$ holds :$s\_n\; =\; \backslash prod\_^n\backslash pi\_i$ The first $\backslash pi\_i$ are 2, 3, 2, 5, 2, 3, 7, ... . In other words, the quotient of two successive superior highly composite numbers is a prime number. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Superior highly composite number」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.